The Wikipedia article about T-joins explains:
Let T be a subset of the vertex set of a graph. An edge set is called a T-join if in the induced subgraph of this edge set, the collection of all the odd-degree vertices is T. "(Note that in a connected graph, a T-join will always exist as, due to the handshaking lemma, |T| will always be even.)
The handshaking lemma is the statement that every finite undirected graph has an even number of vertices with odd degree (also from Wikipedia). I can't make sense of the last sentence in the above quote. In the first sentence, T is an arbitrary subset, so why does |T| have to be even? The handshaking lemma seems to imply that a T-join cannot exist if |T| is odd.
Am I missing something or does the last sentence in the quote need to be rewritten?
Best Answer
The formulation is misleading to put it mildly, if not downright wrong - for the reason you stated: If $|T|$ is odd, no $T$-join can exist (by the handshaking lemma). If $|T|$ is even and the graph is connected, one can construct a $T$-join for this vertex set $T$.
The sentence "$|T|$ will always be even" can at best be interpreted as "No one in their right mind would investigate odd $|T|$ because it is so obvious that no T-joins exist then", but this is not what the quote literally says. If you don't want to edit this, I'll do soon (I already got rid of that stupid quotation mark - another hint to possible qulity problems with that paragraph - before writing down this answer)